Ann. Acad. Rom. Sci.

Ser. Math. Appl.

ISSN 2066-6594

Vol. 18, No. 2/2026

SPECTRAL PROPERTIES OF POSITIVE

LINEAR OPERATORS UNDER LACUNARY

STATISTICALLY RELATIVELY UNIFORM

CONVERGENCE^∗

Pranab Jyoti Dowari^†

Munindra Regon^‡

Binod Chandra Tripathy^§

Communicated by S. Trean¸t˘a

DOI

10.56082/annalsarscimath.2026.2.81

Abstract

This paper introduces the spectral properties of positive linear op-

erators under the framework of lacunary statistically relatively uniform

convergence. We deﬁne the concept of lacunary statistically relatively

uniform and establish inclusion and stability results for sequences of

positive linear operators. Furthermore, we examine the convergence of

spectral radii and provide illustrative examples using classical opera-

tors. This work extends both Korovkin-type approximation theory and

spectral theory into the lacunary statistical relatively uniform setting.

Keywords: statistical convergence, lacunary convergence, spectral ra-

dius, point spectrum.

MSC: 40A05, 60B10, 60B12, 60F17.

^∗Accepted for publication on November 07, 2025

^†pranabdowari@gmail.com, Department of Mathematics, Assistant Professor, Morid-

hal College, Dhemaji-787057, Assam

^‡munindraregon@gmail.com, Department of Mathematics, Research Scholar, Dibru-

garh University, Dibrugarh-786004, Assam

^§tripathybc@yahoo.com, tripathybc@rediffmail.com, tripathybc@gmail.com,

Department of Mathematics, Professor, Tripura University, Agartala-799022, Tripura,

India

81

Spectral properties of positive linear operators

82

1 Introduction

A classical milestone in the study of positive linear operators and their ap-

proximation properties is the well-known Korovkin theorem, which charac-

terizes the uniform convergence of sequences of positive linear operators

through the convergence of their action on test functions [1, 19]. Sub-

sequently, many generalizations of the Korovkin theorem have been ob-

tained by relaxing the mode of convergence, leading to statistical approxi-

mation [2,15,18,22 ], lacunary approximation [16,17], and other extensions.

In this direction, Demirci and Orhan [9] studied statistical relative approxi-

mation on modular spaces. Later, Demirci, Dirik, and Yıldız [10] discussed

approximation through statistical relative uniform convergence of sequences

of functions using the power series method, while Demirci, Orhan, and Ko-

lay [11] examined weighted statistical relative approximation by positive lin-

ear operators.

On the other hand, spectral theory provides a powerful framework for

analyzing the behavior of linear operators through their spectra and spec-

tral radii [7,20]. Spectral properties are not only essential in pure operator

theory but also in approximation processes, stability analysis, and ergodic

theory. In particular, investigating the spectrum of sequences of positive lin-

ear operators under various modes of convergence has emerged as a natural

and signiﬁcant question.

A diﬀerent line of generalization began with the introduction of relatively

uniform convergence, ﬁrst studied by Moore [21] and Chittenden [4 , 5]. In

this mode, convergence is measured relative to a scale function, thus pro-

viding more ﬂexibility compared to ordinary uniform convergence. Later,

statistical convergence, originally introduced by Fast [15] and Steinhaus [22 ],

was extended to function spaces and approximation theory [13,14].

Recently, Demirci and Orhan [8] uniﬁed these two approaches by in-

troducing the concept of statistically relatively uniform convergence. They

applied this new framework to Korovkin-type approximation theorems and

demonstrated that it provides strictly stronger results than both statistical

and relatively uniform convergence.

In this paper, we aim to extend this framework further by investigating

the spectral properties of positive linear operators under lacunary statisti-

cally relatively uniform convergence. The lacunary approach, introduced by

Freedman, Sember, and Raphael [16], and later studied by Fridy [17], re-

ﬁnes statistical convergence by using lacunary intervals. Combining it with

relatively uniform convergence provides a rich setting for both approxima-

tion and operator theory. We deﬁne the notion of the lacunary statistically

P.J. Dowari, M. Regon, B.C. Tripathy

83

relatively uniform spectrum, study spectral inclusion and spectral radius con-

vergence theorems, and illustrate our results with classical operators such

as Bernstein and Baskakov operators.

2 Preliminaries

Let X ⊂ R be a non-empty compact set and denote by C(X) the space of

all real-valued continuous functions on X, which becomes a Banach space

under the supremum norm

kfk_C(X)= sup |f(x)|,

f ∈ C(X).

x∈X

A sequence {T_n} of operators on C(X) is called a sequence of positive

linear operators if each T_n: C(X) → C(X) is linear, i.e. T_n(af + bg) =

aT_n(f) + bT_n(g) for all f, g ∈ C(X) and a, b ∈ R, and positive, i.e. f(x) ≥ 0

on X implies (T_nf)(x) ≥ 0 on X. In classical approximation theory one

studies whether T_n(f) → f uniformly for every f ∈ C(X) as n → ∞; a

standard example is the sequence of Bernstein polynomials B_n(f; x), which

converge uniformly to f ∈ C[0, 1].

2.1

Lacunary statistically relatively uniform convergence

Let θ = (k_r) be a lacunary sequence with k₀= 0, h_r= k_r− k_r−1→ ∞, and

intervals

I_r= (k_r−1, k_r],

r ≥ 1.

For a subset A ⊂ N, the lacunary density of A with respect to θ is deﬁned

as

1

δ_θ(A) = lim

^r→∞h_r

|A ∩ I_r|,

whenever the limit exists.

Let σ : X → (0, ∞) be a continuous scale function and endow C(X)

with the weighted sup–norm

|f(x)|

kfk_σ= sup

.

σ(x)

x∈X

Deﬁnition 1. A sequence (f_n) in C(X) is said to converge to f ∈ C(X)

in the lacunary statistically relatively uniform sense (abbreviated lac-SRU,

with respect to σ) if for every ε > 0,

ꢀ

ꢁ

δ_θ{n ∈ N : kf_n− fk_σ≥ ε} = 0.

Spectral properties of positive linear operators

84

We denote this convergence by

lac-st

f_n===⇒ f

(X; σ).

θ

This deﬁnition reduces to ordinary relative uniform convergence when θ

is the trivial sequence k_r= r, and to lacunary statistical convergence when

σ ≡ 1.

2.2

Basic properties

The following properties are straightforward consequences of the deﬁnition.

lac-st

Proposition 1 (Linearity). If f_n===⇒ f and g_n===⇒ g in C(X), then for

θ

all scalars a, b ∈ C,

lac-st

af_n+ bg_n===⇒ af + bg.

θ

lac-st

Proposition 2 (Domination). If f_n===⇒ f and (g_n) is a sequence in C(X)

θ

lac-st

such that kg_n− f_nk_σ→ 0 uniformly in n, then g_n===⇒ f.

θ

Proposition 3 (Implication from uniform convergence). If f_n→ f uni-

lac-st

formly in k · k_σ, then f_n===⇒ f for every lacunary sequence θ.

θ

Proposition 4 (Non-converse). Lac-SRU convergence does not imply uni-

form convergence.

The following example illustrates the above.

Example 1. Consider the sequence of functions

f_n(x) = sin(nx),

x ∈ [0, 2π],

with the constant scale function σ(x) ≡ 1.

First observe that (f_n) does not converge uniformly on [0, 2π]. Indeed, if

there were a uniform limit f, then for each x we would need

lim sin(nx) = f(x).

n→∞

However, the sequence sin(nx) oscillates between −1 and 1 for most x, and

no pointwise limit exists except possibly at special points (e.g., x = kπ).

In particular, since sup_x∈[0,2π]|f_n(x) − f_m(x)| = 2 for inﬁnitely many pairs

(n, m), uniform convergence fails.

P.J. Dowari, M. Regon, B.C. Tripathy

85

On the other hand, lacunary statistical relative uniform (Lac-SRU) con-

vergence may still hold. Choose a lacunary sequence (k_r), for instance

k_r= 2^r. By lacunary averaging, we examine the means

X

1

|f_n(x)|,

I_r= (k_r−1, k_r],

h_r

n∈I_r

where h_r= k_r−k_r−1. Since sin(nx) is oscillatory and symmetric, its average

value over such lacunary intervals tends to zero for each x. Consequently,

we obtain

f_n(x) −−−−−→ 0 on [0, 2π].

Lac-SRU

Thus, (f_n) is Lac-SRU convergent to the zero function, but not uniformly

convergent. This shows that Lac-SRU convergence does not imply uniform

convergence.

These results establish lac-SRU convergence as a natural weakening of

uniform convergence, while still retaining enough structure (linearity, sta-

bility) to allow spectral analysis of operator sequences.

2.3

Operators acting on C(X)

In our setting, we apply lac-SRU convergence to operator sequences. That

is, for a sequence {T_n} of positive linear operators on C(X), we say

lac-st

T_n(f) ===⇒ f

(X; σ),

∀f ∈ C(X),

θ

whenever for every ε > 0,

ꢂ

n

o

1

|T_k(f)(x) − f(x)|

lim

^r→∞h_r

k ∈ I_r: sup

≥ ε

= 0.

σ(x)

x∈X

This generalizes the usual notion of approximation by positive linear

operators in uniform or statistical sense. In particular:

• If σ(x) ≡ 1, we recover lacunary statistical uniform convergence.

• If θ = (n) (i.e., no lacunarity), we recover ordinary statistically rela-

tively uniform convergence.

• If θ = (n) and σ(x) ≡ 1, we recover ordinary statistical uniform con-

vergence.

Spectral properties of positive linear operators

86

3 Main results

lac-st

Theorem 1 (Spectral Inclusion). If T_n(f) ===⇒ f (X; σ) for all f ∈ C(X),

θ

then

lim sup σ(T_n) ⊆ σ(I) = {1}.

n→∞,θ

Proof. Let X be compact and let σ ∈ C(X) satisfy σ(x) > 0 for all x ∈ X.

On C(X) consider the (equivalent) weighted norm

|f(x)|

kfk_σ:= sup

.

σ(x)

x∈X

(The equivalence with the usual sup–norm follows from compactness of X

and continuity/positivity of σ, so 0 < m ≤ σ ≤ M < ∞.)

Write S_n:= T_n− I. The hypothesis says: for every f ∈ C(X) and every

ε > 0,

ꢂ

ꢃ

1

lim

^r→∞h_r

k ∈ I_r: kS_kfk_σ≥ ε

= 0.

(1)

First we show the equiboundedness of (T_n) in k · k_σ).

It is well known that for a positive linear operator T on C(X) one has

kTk_∞→∞= kT1k_∞,

where 1 denotes the constant function 1(x) = 1.

lac-st

By assumption, T_n(1) ===⇒ 1 in the relatively uniform sense. Hence,

θ

there exists r₀∈ N such that for all r ≥ r₀, and for all but a subset of I_r

with lacunary density tending to zero, we have

kT_k1 − 1k_∞≤ 1,

(k ∈ I_r).

This inequality immediately gives,

kT_k1k_∞≤ 2,

for lacunarily many k.

Consequently,

kT_kk_∞→∞≤ 2

for all k in a set of lacunary density 1.

Since the norms k · k_∞and k · k_σare equivalent on C(X) (because σ

is continuous and strictly positive on compact X), there exists a constant

C ≥ 1 such that

kT_kk_σ→σ≤ C

for k in a set of lacunary density 1.

P.J. Dowari, M. Regon, B.C. Tripathy

87

Finally, by discarding a subset of indices of lacunary density zero, we

may assume without loss of generality that

sup kT_kk_σ→σ≤ C < ∞.

(2)

k

Next, we show the transition from point-wise lacunary RU convergence to

operator–norm lacunary statistically RU convergence.

Fix ε > 0. Let B := {f ∈ C(X) : kfk_σ≤ 1} be the unit ball in

(C(X), k · k_σ). Choose a ﬁnite ε/(4C)–net {f⁽¹⁾, . . . , f^(N)} ⊂ B (existence

follows from separability of C(X)). For each j, by (1) there is r_jsuch that

for all r ≥ r_j,

ꢂ

ꢃ

1

k ∈ I_r: kS_kf^(j)k_σ≥ ε/2

≤ ε/2.

h_r

|E_r|

Let r_∗:= max_jr_j. Then for r ≥ r_∗, outside a subset E_r⊂ I_rwith

≤ ε/2,

h_r

we have kS_kf^(j)k_σ< ε/2 for all j = 1, . . . , N.

Now take any f ∈ B and pick j with kf − f^(j)k_σ≤ ε/(4C). For k ∈

I_r\ E_r, using linearity and (2),

kS_kfk_σ≤ kS_kf^(j)k_σ+ kS_k(f − f^(j))k_σ

ε

C + 1

≤

+ (kT_kk_σ→σ+ 1) ·

≤

+

ε ≤ ε.

2

4C

2

4C

Therefore,

ꢂ

ꢃ

1

k ∈ I_r: kS_kk_σ→σ≥ ε

ꢂ

h_r

ꢂ

ꢃ

1

|E_r|

≤

ε

ꢂ

≤

k ∈ I_r: ∃f ∈ B, kS_kfk_σ≥ ε

≤

.

h_r

2

Since ε > 0 was arbitrary, we conclude

ꢂ

ꢃ

1

lim

^r→∞h_r

k ∈ I_r: kT_k− Ik_σ→σ≥ ε

= 0.

(3)

That is, T_k→ I in operator norm in the lacunary statistical sense.

Further, we check for the resolvent invertibility via Neumann series for

λ = 1. Fix λ ∈ C with λ = 1 and set δ := |1 − λ| > 0. By (3 ), for any

η ∈ (0, δ) the set

ꢃ

G_η:= k : kT_k− Ik_σ→σ< η

has lacunary density 1. For k ∈ G_η, write

h

i

T_k− I

T_k− λI = (1 − λ) I +

.

1 − λ

Spectral properties of positive linear operators

88

ꢄ

T_k−I

Since

< η/δ < 1, the bracket is invertible by the Neumann series,

1−λ

σ→σ

hence T_k− λI is invertible. Thus, for every λ = 1, the set {k : λ ∈ σ(T_k)}

has lacunary density 0.

By the previous steps, no λ = 1 can belong to the lacunary lim sup of

the spectra. Since σ(I) = {1}, we obtain

lim sup σ(T_n) ⊆ σ(I) = {1},

n→∞,θ

as claimed.

Theorem 2 (Spectral Radius Convergence). If (T_n) is a sequence of positive

lac-st

linear operators such that T_n(f) ===⇒ f for all f ∈ C(X), then

θ

X

1

lim

r(T_k) = 1.

^r→∞h_r

k∈I_r

Proof. Let X be compact, σ > 0 continuous, and equip C(X) with the

equivalent weighted sup–norm

|f(x)|

kfk_σ:= sup

.

σ(x)

x∈X

As in the proof of the Spectral Inclusion Theorem 3.1, the assumption

lac-st

T_n(f) ===⇒ f

(f ∈ C(X))

θ

implies that

ꢂ

1

ꢂ

lim

{k ∈ I : kT − Ik

≥ ε} = 0,

(4)

ꢂ

r

k

σ→σ

^r→∞h_r

for every ε > 0. That is, T_k→ I in operator norm in the lacunary statistical

sense.

For any bounded operator S, the spectral radius satisﬁes

r(S) ≤ kSk.

Therefore,

|r(T_k) − 1| ≤ kT_k− Ik_σ→σ

,

since r(I) = 1 and r(·) is 1-Lipschitz with respect to operator norm pertur-

bations.

P.J. Dowari, M. Regon, B.C. Tripathy

89

Fix ε > 0 and for suﬃciently large r,

ꢂ

1

ꢂ

{k ∈ I : |r(T ) − 1| ≥ ε} ≤

{k ∈ I : kT − Ik

≥ ε} < ε.

ꢂ

r

k

r

k

σ→σ

h_r

Hence, for large r,

ꢂ

X

1

r(T ) − 1 ≤

|r(T_k) − 1|

ꢂ

k

ꢂ

h_r

k∈I_r

≤ ε · 2 + ε,

since on at most εh_rindices we may have deviation up to 2 (boundedness

from equiboundedness of T_k), and on the rest |r(T_k) − 1| < ε. Thus the

whole average deviation is bounded by 3ε.

As ε > 0 is arbitrary, it follows that

X

1

lim

r(T_k) = 1.

^r→∞h_r

k∈I_r

Theorem 3 (Point Spectrum Approximation). Suppose each T_nadmits an

lac-st

eigenvalue λ_nwith eigenfunction f_n. If f_n===⇒ f = 0, then λ_n→ 1 in

θ

lac-SRU sense.

Proof. Let (T_n) be a sequence of positive linear operators on C(X). By

assumption, for each n there exists λ_n∈ C and f_n∈ C(X) \ {0} such that

T_nf_n= λ_nf_n.

We are also given that

lac-st

f_n===⇒ f = 0,

θ

in the relatively uniform sense with respect to σ. That is, for every ε > 0,

ꢂ

n

o

1

lim

^r→∞h_r

k ∈ I_r: kf_k− fk_σ≥ ε

= 0.

lac-st

First we start with the approximation property of T_n. Since T_n(g) ===⇒ g

θ

for all g ∈ C(X), applying this with g = f gives

lac-st

T_nf ===⇒ f.

θ

Spectral properties of positive linear operators

90

Now, for any k ∈ I_r,

kT_kf_k− T_kfk_σ≤ kT_kk_σ→σkf_k− fk_σ.

From the Spectral Inclusion Theorem, the operators are equibounded in

k · k_σ, say kT_kk_σ→σ≤ C for lacunarily many k. Thus, whenever kf_k− fk_σ

is small, so is kT_kf_k− T_kfk_σ.

But T_kf_k= λ_kf_k, so

λ_kf_k− f = (T_kf_k− T_kf) + (T_kf − f).

Taking k · k_σ, and using the triangle inequality,

kλ_kf_k− fk_σ≤ kT_kf_k− T_kfk_σ+ kT_kf − fk_σ.

By approximation of T_n, the second term tends to 0 lacunary statistically.

The ﬁrst term tends to 0 lacunary statistically as well (since f_k→ f lac-st

RU and {T_k} is equibounded). Hence,

lac-st

λ_kf_k===⇒ f.

θ

lac-st

On the other hand, f_k===⇒ f. Therefore, both λ_kf_kand f_kconverge to

θ

the same nonzero limit f in the lac-SRU sense.

Thus, for every ε > 0,

ꢂ

n

o

1

lim

^r→∞h_r

k ∈ I_r: kλ_kf_k− f_kk_σ≥ ε

= 0.

We see that,

kλ_kf_k− f_kk_σ= |λ_k− 1| kf_kk_σ.

lac-st

Since f_k===⇒ f = 0, there exists δ > 0 and a set of lacunary density 1 such

θ

that kf_kk_σ≥ δ for all those k.

Therefore, on a set of indices of lacunary density 1 we have

kλ_kf_k− f_kk_σ

|λ_k− 1| ≤

.

kf_kk_σ

But the numerator converges to 0 in lac-SRU sense, and the denominator

stays bounded away from zero on a density–one subsequence. Hence

λ_k→ 1 in lacunary statistically relatively uniform sense.

P.J. Dowari, M. Regon, B.C. Tripathy

91

4 Examples

We illustrate the abstract results with some classical positive linear operators

from approximation theory. These operators are well-known to approximate

the identity in the sense of Korovkin’s theorem, and we show that they also

exhibit the spectral properties described in the preceding theorems which

we studied under lacunary statistically relatively uniformly convergent.

• Bernstein operators. For f ∈ C[0, 1] and x ∈ [0, 1], the n-th Bern-

stein polynomial is deﬁned by

ꢇ ꢈ

n

X

ꢅ ꢆ

n

k

x^k(1 − x)^n−k

.

k

n

(B_nf)(x) =

f

k=0

It is classical that B_nf → f uniformly on [0, 1] for every f ∈ C[0, 1].

Consequently, along any lacunary sequence θ = (k_r) we have

lac-st

B_nf ===⇒ f,

θ

so Bernstein operators satisfy the lac-SRU convergence. Spectrally,

since B_n(1) = 1, we have 1 ∈ σ(B_n) for each n, and the spectral radius

satisﬁes r(B_n) = 1. By the Spectral Radius Convergence Theorem, we

recover

X

1

lim

r(B_k) = 1.

^r→∞h_r

k∈I_r

• Baskakov operators. On C[0, ∞), the n-th Baskakov operator is

given by

ꢇ

ꢈ ꢇ

ꢈ

∞

k

n

X

ꢅ ꢆ

n + k − 1

x

1

k

n

(V_nf)(x) =

f

.

k

1 + x

k=0

These are positive linear operators satisfying V_n(1) = 1 and V_n(t) = t

for the test functions 1 and t. By Korovkin’s theorem, V_nf → f

uniformly on compacts of [0, ∞). Thus, in the lacunary statistical

sense with respect to σ(x) = 1 + x, we have

lac-st

V_nf ===⇒ f,

f ∈ C[0, ∞).

θ

As in the Bernstein case, the constant function 1 is an eigenfunction

with eigenvalue 1, and the point spectrum accumulates at {1}. Hence

the lac-SRU spectrum reduces to {1}.

Spectral properties of positive linear operators

92

• Sz´asz–Mirakyan operators. On C[0, ∞), the Sz´asz–Mirakyan op-

erators are deﬁned by

∞

k

X

ꢅ ꢆ

(nx)

(S_nf)(x) = e^−nx

f

.

k

n

k!

k=0

They satisfy S_n(1) = 1 and S_n(t) = t, and again by Korovkin’s the-

orem we have S_nf → f uniformly on compacts. Therefore, for the

weight σ(x) = 1 + x,

lac-st

S_nf ===⇒ f,

f ∈ C[0, ∞).

θ

Since S_nis positive and S_n(1) = 1, the eigenvalue 1 persists for each

n, and by the Point Spectrum Approximation Theorem, any other

eigenvalue must converge to 1 in the lac-SRU sense. Hence the limiting

point spectrum is trivial and consists of {1}.

5 Conclusion

We introduced the notion of the lacunary statistically relatively uniform

spectrum of positive linear operators and established spectral inclusion and

stability results. This provides a new connection between approximation

theory and spectral theory in the lacunary framework. Future research may

extend these results to uncertain normed spaces, double lacunary sequences,

or pseudospectral analysis.

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